The concept of spinor is now important in theoretical physics but
it is a difficult topic to gain acquaintance with. Spinors were defined
by Elie Cartan, the French mathematician, in terms of three
dimensional
vectors whose components are complex. The vectors which are of
interest
are the ones such that their dot product with themselves is zero.

Let X=(x1, x2, x3) be an element of
the vector space C3. The dot product of X with itself,
X·X, is (x1x1+x2x2+x3x3.
Note that if x=a+ib then x·x=x2=a2+b2 + i(2ab),
rather that a2+b2, which is x times the conjugate of x.

A vector X is said to be isotropic if X·X=0. Isotropic
vectors could be said to be orthogonal to themselves, but that terminology
causes mental distress.

It can be shown that the set of isotropic vectors in C3 form
a two dimensional surface. This two dimensional surface can be
parameterized
by two coordinates, z0 and z1 where

z0 = [(x1-ix2)/2]1/2
z1 = i[(x1+ix2)/2]1/2.

The complex two dimensional vector Z=(z0, z1)
Cartan calls a spinor. But a spinor is not just a two dimensional
complex vector; it is a representation of an isotropic three dimensional
complex vector. A vector in C2 has associated with it
the isotropic vector

x1 = z02 - z12
x2 = i(z02 + z12)
x3 = -z0z1.

For any isotropic vector in C3 there will be two vectors in
C2, corresponding to X; i.e., (z0, z1) and
(-z0, -z1). Both of these will map into the same
isotropic X.

When operations such as rotations are carried out on the isotropic
vectors the results in terms of the spinor representations are quite
interesting. For example, suppose X = (1, i, 0). This is an isotropic
vector and its spinors are Z=(1,0) and Z=(-1,0). If X is rotated
about
the x3 axis through an angle θ it becomes
(cos(θ)-isin(θ), sin(θ)+icos(θ, 0).
This is the same as

(exp(-iθ), iexp(-iθ), 0) = exp(-iθ)(1, i, 0) = e-iθX.

The components of the spinor for X become

z0 = [(exp(-iθ) - i(iexp(-iθ)))/2]1/2 = e-iθ/2 and z1=0.

Thus Z becomes exp-iθ/2Z, a rotation of θ/2.

When X is rotated through an angle 2π the spinors for X
get rotated through an angle of π and thus Z goes to -Z. It takes
a rotation of 4π of the isotropic vector to rotate Z back to Z.

It is impossible to visual depict isotropic vectors and spinors because
three dimensional complex vectors involve six dimensions and spinors as two
dimensional complex vectors involve four dimensions.

For other interesting properties of vectors with complex components see
Bezout's Theorem.